Same problem here. I have ArcGIS for personal use. When I login to AGOL, I get the same message box as in the original post that hangs. And, there's a popup in the upper right of my browser window (Chrome) that says "Unable to apply this update automatically".
I have also tried different browsers, cleared my browser history, cookies, cache, etc., restarted my computer, and unplugged and plugged it back in (joking on the last one). No change...
My company has separate license and there have been no issues with those accounts. So maybe this is just limited to the personal use accounts.
Please post if there is resolution to this. Hopefully, that will also fix whatever I'm experiencing too.
Thanks